Detection of MPSK, including quaternary phase-shift keying or QPSK, is accomplished beginning with complex synchronous demodulation in which real (R) and imaginary (I) components of the MPSK are each synchronously detected. Thereafter, the MPSK receiver performs a symbol detection procedure. In order to synchronously demodulate the received signal, the oscillation frequency of the local carrier wave in the receiver has to match the frequency of the modulated carrier wave and no phase error should be present. As a result, the original signal can be perfectly detected by synchronous demodulation. To achieve this, a carrier wave generated within the receiver by a local oscillator (or beat-frequency oscillator) is synchronized with the suppressed carrier of the modulated signal. The conventional technology for synchronizing the carrier wave generated by the local oscillator with the suppressed carrier of the received signal will be described, referring to FIGS. 1, 2A and 2B.
FIG. 1 is a block diagram showing an automatic frequency control apparatus for conventional quadrature phase-shift keying (QPSK) demodulation. In FIG. 1, the received signal S(t) is supplied to multipliers 11 and 12. Multipliers 11 and 12 multiply the received signals by first and second local carrier waves which are supplied from a voltage-controlled oscillator (VCO) 19 and differ from each other in phase by 90.degree.. Here, the first and second local carrier waves can be represented as 2cos .omega..sub.1 t and 2sin .omega..sub.1 t, respectively.
The output signals of the multipliers 11 and 12 are low-pass filtered by low-pass filters (LPF) 13 and 14, respectively, to become baseband signals. When Gaussian noise is not included in the received signal, the received signal S(t) can be expressed as Acos[.omega..sub.0 t+.phi.(t)]. The low-pass filter 13 responds to the output signal of the multiplier 11 to supply an I-channel (in-phase) signal I(t) expressed as Acos[.DELTA..omega.t+.phi.(t)], and the low-pass filter 14 responds to the output signal of the multiplier 12 to supply a Q-channel (quadrature) signal Q(t) expressed as Asin[.DELTA..omega.t+.phi.(t)]. Here, .DELTA..omega.=.omega..sub.1 -.omega..sub.0.
The responses of the low-pass filters 13 and 14 are digitized in analog-to-digital (A/D) converters 15 and 16, respectively, and the digitized signals are supplied as demodulated signals (I.sub.k and Q.sub.k) to a frequency detector 17. The signals supplied from the A/D converters 15 and 16 each have a phase .phi.(t) that can change every symbol period T.sub.b. For example, in quadrature phase-shift keying, phase .phi.(t) is 45.degree., 135.degree., -45.degree. or -135.degree. and varies according to the bit stream information supplied for transmission, the variation having a symbol period T.sub.b. The frequency detector 17 generates frequency offset information V(k) responsive to the two channel signals, I.sub.k and Q.sub.k, that it receives as input signals. The frequency offset is generated when the actual oscillation frequency (local carrier wave) of the local oscillator (or beat-frequency oscillator) of a receiver and the frequency of the received carrier wave do not coincide. The frequency offset information V(k) is supplied through a loop filter 18 to a control input of the VCO 19, which generates the first and second local carrier waves according to the input frequency offset information. The first and second local carrier waves have the same frequency and differ in phase by 90.degree., for generating the I- and Q-channel signals. The technology for detecting the above-described frequency offset information is discussed in "AFC Tracking Algorithms," IEEE Transactions on Communications, Vol. COM-32, No. 8, pp.935-947.
FIG. 2A shows a frequency detector using the cross product method for generating V(k) disclosed in the above reference. The I-channel signal I.sub.k, which is sampled and digitized, is applied to a delay circuit (D) 21 and a multiplier 24, and the Q-channel signal Q.sub.k, which is also sampled and digitized, is applied to a delay circuit 22 and a multiplier 23. The multiplier 23 multiplies the delayed I-channel signal by the input Q-channel signal, and the multiplier 24 multiplies the delayed Q-channel signal by the input I-channel signal. A subtractor 25 subtracts the output signal of multiplier 24 from the output signal of multiplier 23 and generates frequency offset information V(k) which is determined according to a sampling period T.sub.s and symbol period T.sub.b.
That is, when sampling period T.sub.s is the same as symbol period T.sub.b, i.e., T.sub.s =T.sub.b, the subtractor 25 generates frequency offset information V(k) in accordance with the following. ##EQU1## Here, it is assumed that .DELTA..omega.=.omega..sub.1 -.omega..sub.0, .phi.(t)=.phi..sub.k, kT.sub.5 .ltoreq.t&lt;(k+1)T.sub.s and .theta..sub.k =.phi..sub.k -.phi..sub.k-1.
On the other hand, when T.sub.s &lt;T.sub.b (over-sampling: T.sub.b =nT.sub.s, where n&gt;1), frequency offset information V(k) generated from the subtractor 25 is expressed per the formula (20) which follows later on.
Here, when k is not equal to nl, then ##EQU2## and when k is equal to nl, then ##EQU3## wherein l=k/n and l is an integer not over [x]=x.
FIG. 2B shows a frequency detector that uses an arc tangent operation. An arc tangent portion 27 of the FIG. 2B frequency detector receives the signals (Q.sub.k and I.sub.k) of two channels and performs the arc tangent operation, utilizing the Q-channel signal Q.sub.k as a numerator and the I-channel signal I.sub.k as a denominator. The output signal of arc tangent portion 27 is supplied to a differentiator 28 which generates frequency offset information V(k) expressed by formulas (21) and (22).
First, from formulas (1) and (3), F(k) is defined as follows when ##EQU4##
In the case of over-sampling and k.noteq.nl, from formulas (8) and (10), one gets ##EQU5## and from formulas (9) and (11), one gets ##EQU6##
On the other hand, in over-sampling where k=nl, from formulas (14) and (16), one gets ##EQU7## and from formulas (15) and (17), one gets ##EQU8## wherein l=k/n and l is an integer not over [x]=x.
The above formula (21) represents the output signal of differentiator 28 in the case where T.sub.s is equal to T.sub.b, and formula (22) represents the output signal of differentiator 28 in the case of over-sampling, that is, T.sub.s &lt;T.sub.b and T.sub.b =nT.sub.s. Terms .theta..sub.k and .theta..sub.l, the values of which vary according to the transmitted information, are excluded from the formula, in order to detect the exact frequency. Here, .theta..sub.k and .theta..sub.l represent the difference between the phase information transmitted from the current symbol period and the phase information transmitted from immediately preceding symbol period, that is, the phase difference between symbols during transmission. Hereinafter, .theta..sub.k is referred to as transmission phase information. A QPSK demodulator having a phase difference detector for performing an arc tangent operation as above and performing an AFC function is disclosed in European Patent Application No. 0 526 836 A2.
However, as shown in formulas (7) and (21), transmission phase information .theta..sub.k is included in the formula in the case where the sampling frequency is equal to the symbol rate. Thus, it is impossible to detect the exact frequency directly proportional to frequency offset .DELTA..omega.. Also, in the case of over-sampling, as shown in formulas (20) and (22), it is impossible to detect the exact frequency, since transmission phase information .theta..sub.l remains in a sample which undergoes symbol transition. Although the frequency detection performance can be improved by increasing the degree of over-sampling, the cost of hardware for over-sampling is increased for symbol rates above 20 MHz, as used in a direct broadcasting satellite. This problem can be also generated in a differentiator AFC circuit or a discrete Fourier transform AFC circuit.
Accordingly, the inventor sought an automatic frequency control method obtaining a reference phase which is the closest to the phase of a transmitted signal, using the quantization characteristic of the phase during transmission and the phase difference between the most current complex-number sample of the received MPSK signal and the complex-number sample a symbol period earlier, in effect sampling exactly at the Nyquist rate for symbols. The inventor also sought to develop an automatic frequency control apparatus suitable for performing that method.